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Formalizing non-standard arguments in second-order arithmetic

Published online by Cambridge University Press:  12 March 2014

Keita Yokoyama
Keita Yokoyama*
Affiliation:
Mathematical Institute, Tohoku University, Aoba, Sendai, 980-8578, Japan. E-mail: [email protected]
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Abstract

In this paper, we introduce the systems ns-ACA0 and ns-WKL0 of non-standard second-order arithmetic in which we can formalize non-standard arguments in ACA0 and WKL0, respectively. Then, we give direct transformations from non-standard proofs in ns-ACA0 or ns-WKL0 into proofs in ACA0 or WKL0.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 75 , Issue 4 , December 2010 , pp. 1199 - 1210
Copyright
Copyright © Association for Symbolic Logic 2010

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References

REFERENCES

[1]Avigad, Jeremy, Formalizing forcing arguments in subsystems of second-order arithmetic, Annals of Pure and Applied Logic, vol. 82 (1996), pp. 165–191.CrossRefGoogle Scholar
[2]Buss, Samuel R. (editor), Handbook of proof theory, Studies in Logic and the Foundations of Mathematics, vol. 137, North-Holland, 1998.Google Scholar
[3]Enayat, Ali, From bounded arithmetic to second order arithmetic via automorphisms, Logic in Tehran (Enayat, Ali, Kalantari, Iraj, and Moniri, Mojtaba, editors), Lecture Notes in Logic, vol. 26, ASL and A. K. Peters, 2006.CrossRefGoogle Scholar
[4]Hajek, P. and Pudlak, P., Metamathematics of first-order arithmetic, Springer, Berlin, 1993.CrossRefGoogle Scholar
[5]Kaye, R., Models of Peano Arithmetic, Oxford University Press, 1991.CrossRefGoogle Scholar
[6]Keisler, H. Jerome, Nonstandard arithmetic and reverse mathematics, The Bulletin of Symbolic Logic, vol. 12 (2006), no. 1, pp. 100–125.CrossRefGoogle Scholar
[7]Sakamoto, N. and Yokoyama, K., The Jordan curve theorem and the Schönflies theorem in weak second-order arithmetic, Archive for Mathematical Logic, vol. 46 (2007), pp. 465–480.CrossRefGoogle Scholar
[8]Simpson, Stephen G., Subsystems of second order arithmetic, Perspectives in Mathematical Logic, Springer-Verlag, 1999, second edition, Perspectives in Logic, Association for Symbolic Logic, Cambridge University Press, 2009.CrossRefGoogle Scholar
[9]Tanaka, K. and Yamazaki, T., A non-standard construction of Haar measure and weak König's lemma, this Journal, vol. 65 (2000), no. 1, pp. 173–186.Google Scholar
[10]Tanaka, Kazuyuki, Non-standard analysis in WKL0, Mathematical Logic Quarterly, vol. 43 (1997), no. 3, pp. 396–400.CrossRefGoogle Scholar
[11]Tanaka, Kazuyuki, The self-embedding theorem of WKL0 and a non-standard method, Annals of Pure and Applied Logic, vol. 84 (1997), pp. 41–49.CrossRefGoogle Scholar
[12]Yokoyama, Keita, Non-standard analysis in ACA0 and Riemann mapping theorem, Mathematical Logic Quarterly, vol. 53 (2007), no. 2, pp. 132–146.CrossRefGoogle Scholar